{
  "id": "2311.09421",
  "version": "v1",
  "published": "2023-11-15T22:38:57.000Z",
  "updated": "2023-11-15T22:38:57.000Z",
  "title": "The Kakimizu complex for genus one hyperbolic knots in the 3-sphere",
  "authors": [
    "Luis G. Valdez-Sánchez"
  ],
  "comment": "56 pages, 28 figures. Comments are welcome",
  "categories": [
    "math.GT"
  ],
  "abstract": "The Kakimizu complex $MS(K)$ for a knot $K\\subset\\mathbb{S}^3$ is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the exterior of $K$ and simplices any set of vertices with mutually disjoint representative surfaces. In this paper we determine the structure of the Kakimizu complex $MS(K)$ of genus one hyperbolic knots $K\\subset\\mathbb{S}^3$. In contrast with the case of hyperbolic knots of higher genus, it is known that the dimension $d$ of $MS(K)$ is universally bounded by $4$, and we show that $MS(K)$ consists of a single $d$-simplex for $d=0,4$ and otherwise of at most two $d$-simplices which intersect in a common $(d-1)$-face. For the cases $1\\leq d\\leq 3$ we also construct infinitely many examples of such knots where $MS(K)$ consists of two $d$-simplices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-15T22:38:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10"
    ],
    "keywords": [
      "hyperbolic knots",
      "kakimizu complex",
      "minimal genus seifert surfaces",
      "isotopy classes",
      "higher genus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}